How to calculate percentages in your head: 10%, 1% and growth factors

A percentage is a fraction with denominator 100: $x\%$ means $\dfrac{x}{100}$. Every percentage problem reduces to a multiplication, and mental math becomes easy once you break the rate into pieces you can do without paper.

The definition that solves every case

To find $x\%$ of a value $V$:

Example: $35\%$ of $80$ is $0.35 \cdot 80 = 28$. This form works on a calculator and for checking. In your head, decomposing is faster.

The anchors: 10%, 5% and 1%

• 10%: divide by 10 (move the decimal point one place). $10\%$ of $470$ is $47$.
• 5%: half of 10%. $5\%$ of $470$ is $23.5$.
• 1%: divide by 100. $1\%$ of $470$ is $4.7$.
• 50% and 25%: half, and half of half.

With these anchors, almost any rate becomes a sum or a difference.

Worked example: 15% of 240

1. $10\%$ of $240$ is $24$.
2. $5\%$ is half of that: $12$.
3. Add: $24 + 12 = 36$.

Worked example: 19% of 300

1. $20\%$ of $300$ is $2 \cdot 30 = 60$.
2. $1\%$ of $300$ is $3$.
3. Subtract: $60 - 3 = 57$.

Swap the order when it helps

Since $\dfrac{x}{100} \cdot y = \dfrac{y}{100} \cdot x$, $x\%$ of $y$ equals $y\%$ of $x$. It looks like a trick, but it saves time: $4\%$ of $75$ is the same as $75\%$ of $4$, which is $3$.

Increases and discounts with a multiplier

Increasing a value by $i\%$ means multiplying by $1 + \dfrac{i}{100}$; discounting $i\%$ means multiplying by $1 - \dfrac{i}{100}$.

Worked example: a $180 item with a 15% discount

1. Discount factor: $1 - 0.15 = 0.85$.
2. Final price: $180 \cdot 0.85 = 153$.
3. Mental check: $15\%$ of $180$ is $18 + 9 = 27$, and $180 - 27 = 153$.

Successive percentages

Successive changes multiply; they do not add. A $10\%$ increase followed by another $10\%$ gives a factor of $1.1 \cdot 1.1 = 1.21$: a total increase of $21\%$, not $20\%$.

Worked example: a 25% increase followed by a 20% discount

1. Factors: $1.25$ and $0.80$.
2. Product: $1.25 \cdot 0.8 = 1$.
3. The price returns exactly to the original.

Finding the rate or the original value

• What percentage of $b$ is $a$? Compute $\dfrac{a}{b} \cdot 100$. For example, $18$ out of $72$ is $\dfrac{18}{72} = 0.25 = 25\%$.
• Value before an increase: if the price after a $20\%$ increase is $96, the original is $\dfrac{96}{1.2} = 80$. A common mistake is to subtract $20\%$ of $96$, which gives $76.80$.

Most common mistakes

1. Adding successive rates instead of multiplying the factors.
2. Taking the percentage of the final value when the question is about the initial one.
3. Confusing percentage points with percent change.